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Unformatted text preview: Investigate How can you model the volume of air in your lungs using a sine function? Note: The instructions that follow assume the use of a plastic bottle to capture exhaled air. If you are using laboratory technology, refer to the instructions that accompany the technology. 1. Time your normal breathing for several minutes, starting when you have fi nished exhaling, and counting the number of complete breaths. Determine the period of your breathing cycle when resting, in seconds. 2. Fill a 2-L plastic bottle with water. Invert the bottle in a sink or basin half full of water. 3. Breathe in normally. Then, exhale through a tube and capture the exhaled air in the bottle. Stop when you reach your normal minimum. Tools 2-L plastic bottle measuring cup or graduated cylinder fl exible tube sink or basin with water grid paper or graphing technology Optional spirometer computer with capture and graphing software such as Logger Pro installed OR graphing calculator with CBL 2 TM Calculator-Based Laboratory Sinusoidal Functions of the Form f ( x ) a sin [ k ( x d )] c and f ( x ) a cos [ k ( x d )] c Many real-world applications can be modelled with sinusoidal functions. However, the construction of the model usually requires one or more transformations to fi t the function to the data. One example is the volume of air contained in your lungs during normal breathing. The air inhaled and exhaled can be measured using a spirometer. In this section, you will learn how to transform the sine and cosine functions so that you can use them as models for real-world applications later in the chapter. 5.3 270 MHR Advanced Functions Chapter 5 4. Refi ll the bottle using a measuring cup to determine the volume of air that you breathed out, in millilitres. 5. The model you will use to represent the volume, V , in millilitres, of air in your lungs, versus time, t , in seconds, has the form V a sin [ k ( t d )] c . a) Use the volume from step 4 to determine the amplitude, a , of the function. b) The value of c is derived from the volume of air that remains in your lungs after a normal exhalation. This is difficult to measure directly. A reasonable estimate for the value of c is to add 2.4 L to your value of a . c) Explain why the value of the phase shift, d , is _ 2 . Refer to step 1. d) Use the period you calculated in step 1 to determine the value of k . e) Write an equation to model the volume of air in your lungs. 6. Graph at least two cycles of the model you constructed in step 5. 7. Reflect Inspect the graph from step 6. Explain how it correctly shows the period, the maximum value, the minimum value, and the phase shift of your breathing cycle....
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- Spring '11